Idempotent States on Locally Compact Quantum Groups

Abstract

Abstract. Idempotent states on a unimodular coamenable locally compact quantum group A are shown to be in one-to-one correspondence with right invariant expected C * -subalgebras of A. Haar idempotents, that is, idempotent states arising as Haar states on compact quantum subgroups of A, are characterised and shown to be invariant under the natural action of the modular element. This leads to the one-to-one correspondence between Haar idempotents on A and right invariant symmetric expected C * -subalgebras of A without the unimodularity assumption. Finally the tools developed in the first part of the paper are applied to show that the coproduct of a coamenable locally compact quantum group restricts to a continuous coaction on each right invariant expected C * -subalgebra.Idempotent probability measures on locally compact groups arise naturally as limit distributions of random walks. By analogy, when one considers quantum random walks in the setup provided by topological quantum groups [FS 1 ], one is led to consider idempotent states on locally compact quantum groups. Since the work of Kawada and Itô [KaI], idempotent probability measures have been well understood, as they all arise as Haar measures on compact subgroups. In the quantum world, as shown in [Pal], the situation is more complicated, as already some finite quantum groups admit idempotent states which cannot be canonically associated with any quantum subgroup. Motivated by this discovery U. Franz and the second-named author, later joined by R. Tomatsu, have begun a systematic investigation of idempotent states on finite and compact quantum groups [FS 2−3 , FST]. In particular, necessary and sufficient conditions for such states to be Haar idempotents, i.e. to arise as Haar states on closed quantum subgroups, have been identified, and close relations to expected right invariant C * -subalgebras (called in [FS 3 ] coidalgebras) uncovered. On the other hand the first-named author, inspired by the harmonic analysis considerations due to Lau and Losert showed in a recent paper [Sa] a one-to-one correspondence between compact quantum subgroups of a coamenable locally compact quantum group A and certain right invariant unital C * -subalgebras of A.Motivated by these developments, in this paper we study idempotent states and related structures on coamenable locally compact quantum groups in the sense of [KuV]. At first it might appear that the algebraic formalism here is similar to that encountered in [FS 2−3 ] and [FST], but technical aspects of the locally compact theory (such as the absence of a natural dense Hopf * -algebra) make the problems we2000 Mathematics Subject Classification. Primary 46L65, Secondary 43A05, 46L30, 60B15.

Idempotent States and the Inner Linearity Property

“…However, in the general quantum case, this fundamental result does not hold [14]. We refer to the series of papers [9], [10], [11], [15] for more details on this question, and for the general theory of idempotent states.…”

“…The aim of the present paper is to develop an analytic point of view on these notions, by relating them to the theory of idempotent states, developed in [9], [10], [11], [15].…”

Relative Fourier transforms and expectations on coideal subalgebras

“…Conversely, if our E : H → A is positive in the sense that (1) holds in A u then it extends to a C * expectation. This type of extensibility is of interest, for instance, in the literature on idempotent states on (locally) compact quantum groups [7,6,8,19,20]. Idempotent states are, just as the name suggests, non-commutative analogues of idempotent measures on classical locally compact groups.…”

“…Since for fixed v ∈ V and w ∈ W (with corresponding v * ∈ V * and w * ∈ W * obtained using the compatible inner products) the right hand sides of (20) and (21) are obtained from one another by applying the * operation of H, it follows that indeed (19) and (21) correspond to each other through the construction…”

Lattice of Idempotent States on a Locally Compact Quantum Group

“…Idempotent states on locally compact quantum groups have been investigated in a number of papers, e.g. [25,8,7,2,28,9,27,29,13,6]. The main idea behind these investigations was based on the classical result of Kawada and Itô which establishes a bijection between idempotent states on a classical locally compact group G and compact subgroups of G with the state given by integration with respect to the Haar measure of the subgroup.…”