amalgamation of inverse semigroups and operator algebras

AMALGAMATION OF INVERSE SEMIGROUPS AND OPERATOR ALGEBRAS by Steven P. Haataja A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of D octor of Philosophy Major: Mathematics Under the Supervision of Professors John C. Meakin and Allan P. Donsig Lincoln, Nebraska August, 2006 UMI Number: 3218333 3218333 2006 UMI Microform Copyright All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 by ProQuest Information and Learning Company. AMALGAMATION OF INVERSE SEMIGROUPS AND OPERATOR ALGEBRAS Steven P. Haataja, Ph.D. University of Nebraska, 2006 Advisors: John C. Meakin and Allan P. Donsig We focus on three constructions: amalgamated free products of inverse semi- groups, C*- algebras of inverse semigroups, and amalgamated free products of C*- algebras. The starting point is an amalgam [S 1 , S 2 , U] of inverse semigroups that is full, i.e., the embeddings of U into S 1 and S 2 are bijective on the semilattice of idempo tents. Although the order structure of the amalgamated free product is well- understood, the structure of the maximal subgroups was somewhat mysterious prior to this work. We use Bass-Serre theory to characterize these maximal subgroups and determine which graphs of groups arise in this setting. We obtain necessary and sufficient conditions fo r the amalgamated free product to have trivial subgroups. One surprising consequence is that an amalgamated free product of finite inverse semigroups may be finite. We analyze the structure of the C*-algebra of an inverse monoid S using techniques developed by Sieben. Let E be the semilattice of idempotents of S, and extend t he Munn action of S on E to a partial action of S on C ∗ (E). We prove that C ∗ (S) is isomorphic to the par t ia l crossed product of C ∗ (E) and S using this action. To generalize our construction to inverse semigroups, we determine the effect on C ∗ (S) of attaching an identity to S. Our construction simplifies the construction given by Paterson. Finally we consider C ∗ (S) when the inverse semigroup S is the amalgamated fr ee product of a full amalgam [S 1 , S 2 , U]. We prove that the C*-functor commutes with the formation of amalgamated free products under this hypothesis. We prove an analogous result for the complex algebra of S. Using the characterization of maximal subgroups given above, we identify some amalgamated free products of C*-algebras by recognizing them as C*-algebras of inverse semigroups. Thus, we can identify certain amalgams whose K-theory was found by McClanahan. iii ACKNOWLEDGEMENTS First I would like to thank everyone who provided much needed assistance in the weeks after my ice-skating accident. I will be forever grateful to Jamie Radcliffe, Wendy Hines, Deanna Turk, Terri Moore, Mu-wan Huang, Ian Pierce, David Milan, Bob Ruyle, Martha Gregg, Marilyn Johnson, and my neighbor Jay Penner for all the times they went out of their way for me during my two months in a wheelchair. To my parents I can hardly express in words my grat itude for all of the love and suppo r t they have given me. Even when my life too k an abrupt turn, they were there for me, either in person or in spirit. I love you. I would like to thank the faculty at Black Hills State University for the enthusiasm for learning I gained from them. They are also primarily responsible for my teaching style, although I probably take enjoying teaching further than they would. The faculty at UNL have been wonderful. With each different course I took, their enthusiasm drew me in, providing me with yet anot her topic I wanted to study in detail. My broad mathematical interests are no doubt due to them. I’d also like to thank the members of my committee for t heir questions, comments, suggestions and nitpicks. My co-advisors John Meakin and Allan Donsig were the worst (ahem! best) nitpickers of them all. It’s tough to learn two fields; but they made it happen. Finally I must thank Tim and Natalie Sorenson, the best friends one could imagine. They have selflessly come through for me over the years. Tim has gone above and beyond the call many times, and it was Natalie who gently pushed me back to UNL to pursue my Ph.D. You two are the best. iv Contents 1 Preliminaries 1 1.1 Semigroups and inverse semigroups . . . . . . . . . . . . . . . . . . . 2 1.2 Green’s relations and examples . . . . . . . . . . . . . . . . . . . . . 6 1.3 Categories and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Free products and amalgamated free products . . . . . . . . . . . . . 16 1.5 Complex algebras and operator theory . . . . . . . . . . . . . . . . . 18 1.6 Basic representation theory . . . . . . . . . . . . . . . . . . . . . . . 25 1.7 C*-algebras of inverse semigroups . . . . . . . . . . . . . . . . . . . . 28 1.8 Essential K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2 Maximal subgroups of full amalgamated free products 34 2.1 The basic idea of Bass-Serre t heory . . . . . . . . . . . . . . . . . . . 34 2.2 Amalgamation of groupoids . . . . . . . . . . . . . . . . . . . . . . . 36 2.3 Structure of the maximal subgroups . . . . . . . . . . . . . . . . . . . 38 2.4 Examples and applications . . . . . . . . . . . . . . . . . . . . . . . . 45 3 Constructions of the C*-algebra of an inverse semigroup 54 3.1 The covaria nce algebra of an inverse monoid . . . . . . . . . . . . . . 54 3.2 Crossed product decompositions of C ∗ (S) . . . . . . . . . . . . . . . . 62 3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4 C*-algebras of inverse semigroups and full amalgams 72 4.1 The complex algebra of a full amalga m . . . . . . . . . . . . . . . . . 72 4.2 The C*-algebra of a full amalgam of inverse semigroups . . . . . . . . 7 5 4.3 Some K-theory calculations . . . . . . . . . . . . . . . . . . . . . . . 77 Bibliography 83 1 Chapter 1 Preliminaries Throughout N, Z and C denote the non-negative integers, the integers and the complex numbers, respectively. All semigroups will be discrete and countable unless otherwise specified. The zero of a semigroup will usually be denoted θ. Semilattices will always be meet semilattices. Function notation will depend upon context. Semigroup theory and operator algebra theory have conflicting traditions regarding placement of the function symbol. Operator algebraists always place their functions o n the left, so the imag e of x under the function (operator) A is denoted A(x) or more simply Ax. In semigroup theory functions are usually placed on the right, so the image of x under the function ϕ is denoted xϕ. However, when the function correspo nds to an a ction, the function symbol is placed on the appropriate side, so left (resp. rig ht) actions get the function placed on the left (resp. right). This allows for efficient symbolism when both left and right actions are present. Hence when discussing inverse semigroups, functions will generally go on the right, and when discussing operator algebras (including C*- algebras of inverse semigroups) functions will be placed on the left. We will remind the reader of this convention as needed. This chapter provides a brief overview of the mathematics that appears in this thesis. We give essential definitions, a minimal number of standard results, and provide basic examples that will reappear in later chapters. The only new material occurs in Section 1.6. There we provide a means of describing representations of inverse semigroups on Hilbert space in terms of homomorphisms of inverse semigroups. 2 1.1 Semigroups and inverse semigroup s This section contains basic results about semigroups with a focus toward results relevant to inverse semigroups. Further discussion about the ideas mentioned here can b e found in ([4], [5]) for semigroups and in ([22], [34]) for inverse semigroups. A semigroup is a set equipped with an associative binary operation. A monoid is a semigroup that possesses an identity. Given a semigroup S, one can always find a smallest monoid that contains S. The smallest monoid containing S is usually denoted S 1 and is defined as follows. If S possesses an identity, S 1 = S. If S does not have an identity, then set S 1 = S ∪ {1}, where 1 is a symbol not in S. The multiplication in S 1 is given by s · t = st if s, t ∈ S and s ·1 = 1· s = s for any s ∈ S 1 . Another way to attach an identity to a semigroup S is simply to add an identity to S, regardless of whether S already has one. The resulting set is denoted here by S • := S . ∪ {1}, with multiplication given as above for S 1 . An idem potent of S is an element e such that e 2 = e. The set of idempotents of a semigroup is usually denoted E(S) or more simply as E, if the associated semigroup is clear from context. Associated to every idempotent e in S there is a subgroup H e of S given by H e = {x ∈ S : ex = x and xy = yx = e, for some y ∈ S}. The subgroups H e of S, where e ∈ E(S), are the maximal subgroups of S. If all the maximal subgroups of S are trivial, then S is said to be combinatorial. An element a of S is regular if there is a b in S such that aba = a. If b also satisfies bab = b, then b is called an inverse of a. Note that if aba = a then bab is an inverse of a, and both ab and ba are idempotents. A semigroup is regular if every element has at least one inverse. A semigroup is an invers e se migroup if every element has a unique inverse. In this case the inverse of an element a ∈ S is written a −1 or occasionally a ∗ . The latter notation occurs frequently in the operator algebra literature as the inverse operation is an involution, that is, it satisfies (a −1 ) −1 = a and (ab) −1 = b −1 a −1 for all a, b ∈ S. We use the notation a −1 to denote the inverse of the element a ∈ S throughout. A small calculation shows that idempotents commute in an inverse semigroup. This leads to a useful characterization of inverse semigroups. Proposition 1.1.1. [4, Theo rem 1 . 17] A s emigro up S is inverse if and only if S is regular and any two idempotents of S commute. Thus if S is an inverse semigroup, E(S) forms a (meet) semilattice. Also for any 3 s ∈ S and e ∈ E(S), both ses −1 and s −1 es are idempotents. An inverse semigroup has a natural partial order given by s ≤ t if there is an idempo tent e such that s = et. Verifying that this is a partial order is a routine calculation. Note that if s ≤ t as above, then s = et = e(tt −1 t) = t(t −1 et), so it is easy to see that s ≤ t if and only if there is a n f ∈ E(S) such that s = tf. In fact, for any s, t ∈ S. s ≤ t ⇔ s = ss −1 t ⇔ s = ts −1 s Example 1.1.2. Let G be a group with identity ε, let n be a fixed positive integer, and let B n (G) = {(i, g, j) : 1 ≤ i, j ≤ n, g ∈ G} ∪ {θ}. Define a multiplication by (i, g, j)(k, h, l) =    (i, gh, l) if j = k θ otherwise for i, j, k, l ∈ {1, . . . , n}, and g, h ∈ G and set (i, g, j)θ = θ(i, g, j) = θ for all triples (i, g, j). The element θ is the zero of B n (G). A semigroup is called a Brandt semigroup if it can b e expressed in this manner. The nonzero idempotents of B n (G) have the form (i, ε, i), and the unique inverse of (i, g, j) is (j, g −1 , i). Thus B n (G) is an inverse semigroup. If G is the trivial group we suppress all mention of the group: B n := B n ({ε}) = {(i, j) : 1 ≤ i, j ≤ n} ∪ {θ}. One can also describe B n (G) using matrices. For 1 ≤ i, j ≤ n and g ∈ G, let E i,j (g) denote the n × n matrix having g in the (i, j) position and zeros elsewhere. L et 0 n denote the n × n zero matrix, and set B n (G) = {E i,j (g) : 1 ≤ i, j ≤ n, g ∈ G} ∪ {0 n }, with usual matrix multiplication as the product. (The product of the entry g ∈ G by 0 is 0.) It is easy to see that B n (G) ∼ = B n (G) by mapping (i, g, j) → E i,j (g) and θ → 0 n . We thus will describe nonzero elements of B n (G) using either ordered triples or matrices, depending upon circumstances. If G is the trivial group, the matrix notation can be simplified. In this case, we let G = {1} and use the notation E i,j instead of E i,j (1). The elements of B n can be viewed as lying in M n (C). One must be careful when discussing subobjects of classes of semigroups. A sub- semigroup of a semigroup S is a nonempty subset T of S closed under multiplication. An inverse subsemigroup of an inverse semigroup S is a subset T of S closed both under multiplication and taking of inverses. A subsemigroup of an inverse semigroup need not be an inverse semigroup. A (inverse) submonoid of a monoid M is a (inverse) subsemigroup of M that possesses the identity of M. 4 Semigroups also possess ideals. A le f t ideal of a semigroup S is a nonempty subset I of S such that SI ⊆ I, where SI = {si : s ∈ S, i ∈ I}. A nonempty subset I o f S is a right ideal if IS ⊆ I. A (two-sided) ideal I is a subset that is both a left ideal and right ideal. Fo r a nonempty subset A o f S, the ideal generated by A is the smallest ideal of S containing A. It is denoted by A. A semigroup is said to be simp l e if it contains no proper ideals. If S has a zero element, then S is 0-simple if the only nontrivial ideal of S is { θ}. For an element a ∈ S, the principal right (resp. left) ideal generated by a is the smallest right (resp. left) ideal of S containing a, namely the set aS 1 (resp. S 1 a). The (two-sided) principal ideal generated by a is likewise S 1 aS 1 . The notation S 1 is employed, as a need not be an element of aS or Sa in general. If S is regular or inverse, the relation aba = a ensures that a is an element of aS a nd Sa. Homomorphisms of semigroups are defined in the usual way: a homomorphism ϕ : S → T is a set function such that for every s, t ∈ S, (st)ϕ = (sϕ) (tϕ). The terms injection, surjection, and isomorphism have their usual meanings: one-to-one homomorphism, onto homomorphism, and bijective homomorphism, respectively. If ϕ : S → T is surjective, then we say that T is a quotient of S and we may write S/ϕ instead of T. It should be noted that semigroup homomorphisms behave quite differently than group homomorphisms. Every group homomorphism ϕ : G → H determines and is determined by a normal subgroup of G, specifically the kernel of ϕ. There is no such analogue in semigroup homomorphisms. Instead semigroup homomorphisms are determined by the notion of a congruence. A congruence on a semigroup S is an equivalence relation ∼ on S such that for s, t, u ∈ S, if s ∼ t then su ∼ tu a nd us ∼ ut. If ∼ is a congruence on S the quotient of S relative to ∼ is written S/ ∼. Here is an example that shows how much congruences can differ from group cosets. Let S be a semigroup such that xy = y for all x, y ∈ S. Let ∼ be any equivalence relation on S. For any x, y, z ∈ S, x ∼ y implies xz = z = yz and zx = x ∼ y = zy, so ∼ is a congruence. Here is a more useful but very “ungrouplike” example. Suppose S is a semigroup and I is a proper ideal of S. L et S/I be the set (S\I)∪{θ}, and define a multiplication · on S/I by s · t =    st if s, t, st ∈ S \ I θ otherwise [...]... Chapters 2 and 4 we consider full amalgams of 0-direct unions of Brandt semigroups 1.5 Complex algebras and operator theory We study the structure of various algebras associated with an inverse semigroup in Chapters 3 and 4 Here we define these algebras and discuss some basic properties of such algebras All functions are written on the left in this and the next section Definition 1.5.1 Let S be an inverse. .. are thus often referred to as the local monoids of the category Here are some categories of interest in this dissertation: 14 Mon = monoids and monoid homomorphisms InvSgp = inverse semigroups and inverse semigroup homomorphisms InvMon = inverse monoids and inverse monoid homomorphisms Gp = groups and group homomorphisms AbGp = abelian groups and group homomorphisms C-Alg = complex algebras and complex... Let S1 and S2 be two semigroups with a zero The 0-direct union of S1 and S2 is the amalgamated free product of S1 and S2 with core U = {0}, where the embeddings ιi : U → Si send 0 to the zero element of Si (i = 1, 2) The resulting semigroup S = S1 ∗U S2 is the union of S1 and S2 with the zero elements identified One can easily generalize this construction to a 0-direct union of a finite number of semigroups. .. which J = D contains an isomorphic copy of the bicyclic as an inverse subsemigroup This fact is a consequence of [4, Theorem 2.54] Example 1.2.7 We now discuss free inverse semigroups Inverse semigroups form a class of algebras of type (2, 1) with the operations (·, −1), and are defined by the identities x(yz) = (xy)z, xx−1 x = x, (x−1 )−1 = x, (xy)−1 = y −1x−1 , and xx−1 yy −1 = yy −1xx−1 Thus by Birkhoff’s... matrix are numbered 0 The identity of S corresponds to the identity matrix It is not hard to see that the matrix of the product (x−i xj )(x−k xl ) corresponds to the product of the matrix of x−i xj and the matrix of x−k xl We now define an important class of inverse semigroup algebras 22 Definition 1.5.6 For an inverse semigroup S and 1 ≤ p < ∞ let ℓp (S) be the set of all functions ϕ : S → C such that... if and only if P1 commutes with P2 A projection Q is a subprojection of P if P Q = QP = Q This yields a partial order on the set of projections on H: Q ≤ P if and only if P Q = QP = Q We mention several other important classes of operators in B(H) An operator U ∈ B(H) is a unitary operator if UU ∗ = U ∗ U = 1 The set of unitary operators on H form a group, and is denoted U(H) An isometry is an operator. .. linear operators on H is denoted B(H) We shall refer to any element of B(H) as simply an operator The norm of an operator A ∈ B(H) is given by A = sup{ Aξ : ξ ≤ 1} If H = Cn the set of bounded linear operators on H is usually denoted Mn = Mn (C), the n by n matrices over C The adjoint of an operator T is the unique operator T ∗ that satisfies T ξ, η = ξ, T ∗η for all ξ, η ∈ H In Mn the adjoint of a matrix... representations of importance here are the Munn representation and representations of inverse semigroups as operators on a Hilbert space We first discuss the Munn representation Let S be an inverse semigroup having semilattice E We denote by TE the set of all isomorphisms between principal ideals 26 of E It is not hard to show that TE is an inverse semigroup, which is called the Munn semigroup of E Define... any P ≤ X ∗ X, yielding the Munn representation of PI(P) We can now define a representation of an inverse semigroup Definition 1.6.4 A representation of an inverse semigroup S is a homomorphism of inverse semigroups π : S → PI(P), where P is a set of commuting projections on a Hilbert space H and H= π(s)H s∈S For a semilattice E, we refer to a representation of E as a representation as projections on H,... Hence λ is a homomorphism, and thus λ ∈ Rep(S) The key property of the algebras associated to a semigroup S is this Proposition 1.7.3 There is a one-to-one correspondence between any two of the following, for any inverse semigroup S: (a) representations of S, (b) *-representations of C(S), (c) *-representations of ℓ1 (S), (d) *-representations of C∗ (S), The C*-algebra of an inverse semigroup has the . free products of inverse semi- groups, C*- algebras of inverse semigroups, and amalgamated free products of C*- algebras. The starting point is an amalgam [S 1 , S 2 , U] of inverse semigroups that is. 1.6. There we provide a means of describing representations of inverse semigroups on Hilbert space in terms of homomorphisms of inverse semigroups. 2 1.1 Semigroups and inverse semigroup s This section. left and right actions are present. Hence when discussing inverse semigroups, functions will generally go on the right, and when discussing operator algebras (including C*- algebras of inverse semigroups)