For each holomorphic microlocal operator its symbol is defined as a holomorphic function which satisfies a growth condition. A differential or micro (=pseudo-) differential operator of infinite order is naturally regarded as a holomorphic microlocal operator and its symbol coincides with the usual total symbol. Some sufficient conditions of invertibility for holomorphic microlocal operators in terms of symbols are given.

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By means of an auxiliary coarser topology we study certain order properties of inductive tensor algebras over nuclear LF-spaces.

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A necessary and sufficient condition for two quasifree states of CCR's (the canonical commutation relations) to yield quasi-equivalent representations is obtained, in the most general setting where states are non gauge-invariant in general and the symplectic form defining CCR might be degenerate (either from the start or after completion relative to the topology induced by states). The criterion consists of the following two conditions: (1) The induced topologies on the test function space are equivalent (2) operators ˜{S} and ˜{S'} on the completed test function space (completion relative to the induced topology, after taking quotient by the kernel of the representation) giving the two point functions S and S' (relative to any common inner product giving rise to the induced topology) are such that ˜{S}^1/2−(˜{S'})^1/2 is in the Hilbert Schmidt class.

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The L_p-space L_p(M, η) for a von Neumann algebra M with reference to its cyclic and separating vector η in the standard representation Hilbert space H of M is constructed either as a subset of H (for 2{≤}p{≤}∞), or as the completion of H (for 1{≤}p{≤}2) with an explicitly defined L_p-norm. The Banach spaces L_p(M, η) for different reference vector η (with the same p) are isomorphic.
Any L_p element has a polar decomposition where the positive part L_p⁺(M, η) is defined to be either the intersection with the positive cone V_η^1/(2p) (for 2{≤}p{≤}∞) or the completion of the positive cone V_η^1/(2p) (for 1{≤}p<2). Any positive element has an interpretation as the (1/p)^th power ω^1/p of an ω∈M_*⁺ with its L_p-norm given by ||ω||^1/p.
Product of an L_p element and an L_q element is explicitly defined as an L_r element with r⁻¹=p⁻¹+q⁻¹ provided that 1{≤}r, and the Hölder inequality is proved.
The L_p-space constructed here is isomorphic to those defined by Haagerup, Hilsum, and Kosaki.
As a corollary, any normal state of M is shown to have one and only one vector representative in the positive cone V_η^α for each α∈[0, 1/4].

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